① Differentiation
Fi+ ite
① Lett 0,
=
Relate variables in equation
②
8i 6 +
=
*
① Draw diagram
=
of
Finding relative minimalmaxima of f(x,y)
Optimization Steps:
*
Chain rule:U
②
Use chain rule & differentiate
③
+
Absolute minimal maxima:
Test every
② Evaluate f(x,y) each point
③
Eversetofsolutions & everyprintwhere
of
one
the
=
=
f
① Find all critical points of
F ryx th e point
equate them
&
0
③ LetA:fxx, B fxy,) fyy
*
*
ortest to the
=
it
critical point:
A c
B2-ACC8,
intergetto useaprintheaudieof
rovefurve
=
on
the gene
Maxima
relative
relative
BIACCO, A
inflexion
of
B2AC so... saddle point/print
Minima
s
Testlimiting values
o. :
BI AC 0
=
o. :
testfails
.:
Cincluding corners)
Find of a given point
①
② Find parametric of curve
*
and unit fugentrector?
③ Directional derivative if.
-
rapidly
A
function increases most
&
of
rapidly
of
-
variable, treatall other variables
as
=()
range "R", then
f(x,y)
dedy, find bounds firstsketch the
to
as
one variable's limits
set
a
=
"X":
Setthe known one as He outer.
find He offer variable'slimits as numbers.
Integrate by reversing He
*
is
① sketch on
a
Any value:
order of integration:
graph the currentbounds and the region enclosed
Co-ordinates
t
r
a
i
n
:
or8meteirnterninNewswere,
gary'sequation
varVebsitsmagnite n Srdrdo
pride
⑭Ifbrands don'itntersect andbondfora fromthe
+
② Find of a given point
equation 1
to get
=
sally equation
2 Let sercost, yeusing, substitute intoequation
A
① Sketch both curves & find points of intersection
② In bounded region, find min & max values of cory
=
circular
c-any-be et
a circle:
of
the bounds
of
& solve for
terms oft
in
intersection.
& find oat
③ If bounds intersect, equatetter to eachotter
=
(usually using y ecline)
(use when shape
Ris)
Double integrals with Polar
*
Polar-cord Area:
Applications:
by them
2 Find value of known bounds in terms of the other variable & rice-versa
*
①letV Va +Vyi V2
③ Dr if. Y
&
function of the other variable
=
find direction thatgives
=
normally
constants & integrate
order ofdady.... is order in which you integrate
* Volume
ofdirection 10611 /cos o
*
How to
rate ofchange
I) dxdy,
=
in the direction
indirection of
least
Rate of change
Area
*
I
*
of
a
to
To integrate with respect
values
diagram i.e. twhat
this
ake
or i n
I
in
region
&
(same as normal)
write the other onesbounds as a function
③ A S) didy
=
or
Note:itshape is
A SJ rdrdo
=
symmetric
about an
axis/quadrant, can also
half & ten double
of the
splititand find area
it
Surface Area
*
①letdS
volume ofrevolution:
*
getintegral bounds
to
as in area
① Follow some steps
V
about
② Ifrotated ac-axis, S)2+ydA
S)2FcdA
=
Judv=ur-Sudu
IBP:
Quotient rule:
Itmihr
Common integrals:
() T
=
& Find
bounds
③ Sufae
Aren=SdSdA, for 3D shape
=
Ifrelated abouty-axis, V
I
tanc, teasinga
nose
secis-
curves
by graphing
Ssinc--cosse, Sac", St-Intel, Satar-crctant),
Cosk= Sin K
intersection:
a curve
of
Trig identities:
·
·
Sint +cus=1
tan?t+) sec
=
·
& intersection points
Sin 28 2 sinGCost
i.e.
cone,
e
ecs
=
R
projection
=
on
say plane
↓
=
Costconsin
like normal
Use projection to
bounds
get
*
To find volume of3D shape:*
Triple Integrals
given f (x, y,
(1) fx, 3,2)
to
·
Jeen"
In
setup integral:
SSS f(x3,2) dV
Mass:
*
①
Graph all curves/lines in range
G
a
2) dedy
(flexy,
bonds
Volume:SSSdedside
& find
dedydo
foxy, 2)
=
G
steps
2) over 3D range G
Average value of function flex,
①Graph range
Clark ats a-z
as
plane/ec-3/z-y depending
des
usual
Polar co-ordinates triple integrals
shape)
on
② Evaluate triple integral. (or just bxt)
* Use
-
space is symmetric about an axis
when 3D
for cube
ha(x,y)
y, 2):
rdzdrde....
hi (13),
d
x rcuSt, y=rsint, z
*
② Use 3D shape find bounds of
Spherical co-ordinates (r,d,0)
outer
for inner bounds & scy plane for
Since axisymmetric, o8Yz, t is unter, check 2-rplane
Use when 3D
③ Graph zy plane
point
space is symmetric about
for h(r,o) & g(0)
3D
shape &Letx=rsinocoso.dV=rsind drdd de
*
Note:order integration changes based
*
2
to
as
a
y rsin sin O
parallelopiped with 2.x cross-section then graph explane
if
i.e.
rcosd
0 = 0 2+
=
using, 2) rdadet
=
=
Vector field F P(x,y,z)i Q(x,y,z)i R(x,y,z)I
=
y
For sphere:0 0 =
=
·
=
i+ 6i+t is
Grad:grad* if
Vector Fields
or
IP/gro hergrouse,
=
2
be the outer
to
get bounds & y will
*
=
on
of
=
2
=
·
Scalar f(x,y,z)
*
-Identities:00x if 0
② v.(0xF) 0
=
+
+
=
F P(x,3)i Q(x,y);
=
+
=
Operations:->
Divergence
DivF
-
1.F f,P(x,3,z) 6a(x4,2) R(x,y,z)8xx(E) 0fxF f( x,)
+
+
=
(!) (ac
(-)-(-8) i+ (-8) is
Curl-CuLE
F
x x
=
Line
=
x
integrals for vector functions:
For F P(xy,), Q(ex 3, 4, R(14,2)
=
*
same cases as
+
=
given
integrals of
scalar function:
a
& Bounds to
scalar function T f1y,2)
*
=
cationdone:
integrate on
Work done by force
Case 1-Bounds are a curve ( x(H), y(H), 2(H)
=
Find parametric equation
①
② From parametric equation
ofcurveit
(Ie
③ Substitute parametric egs
f(x,y,2).
in
point 1
to point 2. So for to go from
3 to-3,
Independence
*
should
I
·
go frm-3 to 3 if =-t.
or
CCW &bunds
procedure
at
curve
is
2:
normal line integral.
as
of path:
of
S.F. d independent path
if for solar lunction
4,84=E
independentofpath
integration
Test:I If OxF 0, then F x0::
Constant
of
Fis conservative means F is independent path (Can my P,a,R)
=
=
·
are multiple points/lines:
individual curves with
into
-
functions in terms of , y
use
*
① Sketch all the bounds & splitit
Heir own
some
method, but:
-
⑤ Evaluate line integral:(f(x(), y(), z(+) ds dt
Case 2-Rotate Clockwise
=
& same
from paint to
as particle moves dong
I Edrdt (follow
W
④ To getbands for integral, look atwhat"t"value needs to
be to
getfrom
for scalar line integral
J.Edrdt (((p(**) Q(*) +n()]
=
Line
+
=
=
keeping
He parametric equations
② For each individual curve, find
comterdockwise). Note, can only increase
in mind the direction (clockwise
then find as
curve using staffend points,
③ Find bonds for each individed
② Find 4:4
P,8 G,
=
R: d
=
=
8 (* 4(b) -b(a), evaluate dat endpoint
=
L
(pdx (x)
=
+
e
+
↓
-
-
Patstart point
from assignment10, lea 26
vectorfields
I
or
for each curve & sub everything
back into floc, y,2).
Surface integrals of scalarfuctions
Given surface z g(x, y) & scalar faction f(x,y,z)
*
Surface integrals of
** P(x,y,2), Q(x ,1,2), R(1,3,2)
=
=
⑦ (f(x,y,2) dsdt dsdt
Sa
=
cylinder radius
xty r
=
r
by y 0 - > complete the square
=
+
=
xy+2=r-
sphere, radius
yz - - Semisphere
z=
z
Sty2-> Cone
=
zexy
zx
=
-
y2 -
+
double cone
Parabaloid
Sf dsdt..... ISketch bounds
Goodluck
⑦
Common shapes:
ax
x =
+
·
-
=
& draw projection of
surface in
x-y
plane
useprovedondouble integral bandatitfly,
H. A
Given surface 2 f(x, 3), 50 Surface
①
d
it i
④Surface integral Jgef(x,y,94431) isdyde
Putf (x,y)
Note, when calculating
mit vector,
there is upward verter
(kx) & downward vector (k<0). Find the one the question asks for.
=
2 0
=
5 0 (f(x,y) z)
-
=
n=
② Find unitnormal vector
(182)
③ ds 1
=
+
Projectsurface uncery plane find
bounds for double integral
=
*
-
to
⑧ Surface integral
JF.dsdyd
=
Note:i fsurface
*
y fx, 2), swap all
ys & 2, in the process & solve normally.
is
=